\(\int x^3 \cosh ^2(x) \coth ^2(x) \, dx\) [157]

Optimal result

Integrand size = 12, antiderivative size = 102 \[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\frac {3 x^2}{8}-x^3+\frac {3 x^4}{8}-\frac {3 \cosh ^2(x)}{8}-\frac {3}{4} x^2 \cosh ^2(x)-x^3 \coth (x)+3 x^2 \log \left (1-e^{2 x}\right )+3 x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {3 \operatorname {PolyLog}\left (3,e^{2 x}\right )}{2}+\frac {3}{4} x \cosh (x) \sinh (x)+\frac {1}{2} x^3 \cosh (x) \sinh (x) \] Output:

3/8*x^2-x^3+3/8*x^4-3/8*cosh(x)^2-3/4*x^2*cosh(x)^2-x^3*coth(x)+3*x^2*ln(1 
-exp(2*x))+3*x*polylog(2,exp(2*x))-3/2*polylog(3,exp(2*x))+3/4*x*cosh(x)*s 
inh(x)+1/2*x^3*cosh(x)*sinh(x)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.83 \[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\frac {3 x^4}{8}-\frac {3}{16} \left (1+2 x^2\right ) \cosh (2 x)-x^3 \coth (x)+x^2 \left (x+3 \log \left (1-e^{-2 x}\right )\right )-3 x \operatorname {PolyLog}\left (2,e^{-2 x}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 x}\right )+\frac {1}{8} x \left (3+2 x^2\right ) \sinh (2 x) \] Input:

Integrate[x^3*Cosh[x]^2*Coth[x]^2,x]
 

Output:

(3*x^4)/8 - (3*(1 + 2*x^2)*Cosh[2*x])/16 - x^3*Coth[x] + x^2*(x + 3*Log[1 
- E^(-2*x)]) - 3*x*PolyLog[2, E^(-2*x)] - (3*PolyLog[3, E^(-2*x)])/2 + (x* 
(3 + 2*x^2)*Sinh[2*x])/8
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.23, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.583, Rules used = {5973, 3042, 25, 3792, 15, 3042, 3791, 15, 4203, 15, 26, 3042, 26, 4199, 25, 2620, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \cosh ^2(x) \coth ^2(x) \, dx\)

\(\Big \downarrow \) 5973

\(\displaystyle \int x^3 \cosh ^2(x)dx+\int x^3 \coth ^2(x)dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int x^3 \sin \left (i x+\frac {\pi }{2}\right )^2dx+\int -x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx\)

\(\Big \downarrow \) 25

\(\displaystyle \int x^3 \sin \left (i x+\frac {\pi }{2}\right )^2dx-\int x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx\)

\(\Big \downarrow \) 3792

\(\displaystyle \frac {\int x^3dx}{2}-\int x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \int x \cosh ^2(x)dx+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)\)

\(\Big \downarrow \) 15

\(\displaystyle -\int x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \int x \cosh ^2(x)dx+\frac {x^4}{8}+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \int x \sin \left (i x+\frac {\pi }{2}\right )^2dx+\frac {x^4}{8}+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)\)

\(\Big \downarrow \) 3791

\(\displaystyle -\int x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {3}{2} \left (\frac {\int xdx}{2}-\frac {1}{4} \cosh ^2(x)+\frac {1}{2} x \sinh (x) \cosh (x)\right )+\frac {x^4}{8}+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)\)

\(\Big \downarrow \) 15

\(\displaystyle -\int x^3 \tan \left (i x+\frac {\pi }{2}\right )^2dx+\frac {x^4}{8}+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 4203

\(\displaystyle \int x^3dx-3 i \int i x^2 \coth (x)dx+\frac {x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 15

\(\displaystyle -3 i \int i x^2 \coth (x)dx+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 26

\(\displaystyle 3 \int x^2 \coth (x)dx+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle 3 \int -i x^2 \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 26

\(\displaystyle -3 i \int x^2 \tan \left (i x+\frac {\pi }{2}\right )dx+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 4199

\(\displaystyle -3 i \left (2 i \int -\frac {e^{2 x} x^2}{1-e^{2 x}}dx-\frac {i x^3}{3}\right )+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 25

\(\displaystyle -3 i \left (-2 i \int \frac {e^{2 x} x^2}{1-e^{2 x}}dx-\frac {i x^3}{3}\right )+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 2620

\(\displaystyle -3 i \left (-2 i \left (\int x \log \left (1-e^{2 x}\right )dx-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle -3 i \left (-2 i \left (\frac {1}{2} \int \operatorname {PolyLog}\left (2,e^{2 x}\right )dx-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle -3 i \left (-2 i \left (\frac {1}{4} \int e^{-2 x} \operatorname {PolyLog}\left (2,e^{2 x}\right )de^{2 x}-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

\(\Big \downarrow \) 7143

\(\displaystyle -3 i \left (-2 i \left (-\frac {1}{2} x \operatorname {PolyLog}\left (2,e^{2 x}\right )+\frac {\operatorname {PolyLog}\left (3,e^{2 x}\right )}{4}-\frac {1}{2} x^2 \log \left (1-e^{2 x}\right )\right )-\frac {i x^3}{3}\right )+\frac {3 x^4}{8}-x^3 \coth (x)+\frac {1}{2} x^3 \sinh (x) \cosh (x)-\frac {3}{4} x^2 \cosh ^2(x)+\frac {3}{2} \left (\frac {x^2}{4}-\frac {\cosh ^2(x)}{4}+\frac {1}{2} x \sinh (x) \cosh (x)\right )\)

Input:

Int[x^3*Cosh[x]^2*Coth[x]^2,x]
 

Output:

(3*x^4)/8 - (3*x^2*Cosh[x]^2)/4 - x^3*Coth[x] - (3*I)*((-1/3*I)*x^3 - (2*I 
)*(-1/2*(x^2*Log[1 - E^(2*x)]) - (x*PolyLog[2, E^(2*x)])/2 + PolyLog[3, E^ 
(2*x)]/4)) + (x^3*Cosh[x]*Sinh[x])/2 + (3*(x^2/4 - Cosh[x]^2/4 + (x*Cosh[x 
]*Sinh[x])/2))/2
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
 

rule 4199

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_ 
.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp 
[2*I   Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x 
))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 4203

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symb 
ol] :> Simp[b*(c + d*x)^m*((b*Tan[e + f*x])^(n - 1)/(f*(n - 1))), x] + (-Si 
mp[b*d*(m/(f*(n - 1)))   Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1), x] 
, x] - Simp[b^2   Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; Free 
Q[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 0]
 

rule 5973

Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + 
(d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* 
x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] 
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.15

method	result	size
risch	\(\frac {3 x^{4}}{8}+\left (-\frac {3}{32}+\frac {3}{16} x -\frac {3}{16} x^{2}+\frac {1}{8} x^{3}\right ) {\mathrm e}^{2 x}+\left (-\frac {3}{32}-\frac {3}{16} x -\frac {3}{16} x^{2}-\frac {1}{8} x^{3}\right ) {\mathrm e}^{-2 x}-\frac {2 x^{3}}{{\mathrm e}^{2 x}-1}-2 x^{3}+3 x^{2} \ln \left (1+{\mathrm e}^{x}\right )+6 x \operatorname {polylog}\left (2, -{\mathrm e}^{x}\right )-6 \operatorname {polylog}\left (3, -{\mathrm e}^{x}\right )+3 x^{2} \ln \left (1-{\mathrm e}^{x}\right )+6 x \operatorname {polylog}\left (2, {\mathrm e}^{x}\right )-6 \operatorname {polylog}\left (3, {\mathrm e}^{x}\right )\)	\(117\)

Input:

int(x^3*cosh(x)^2*coth(x)^2,x,method=_RETURNVERBOSE)
 

Output:

3/8*x^4+(-3/32+3/16*x-3/16*x^2+1/8*x^3)*exp(x)^2+(-3/32-3/16*x-3/16*x^2-1/ 
8*x^3)/exp(x)^2-2*x^3/(exp(x)^2-1)-2*x^3+3*x^2*ln(1+exp(x))+6*x*polylog(2, 
-exp(x))-6*polylog(3,-exp(x))+3*x^2*ln(1-exp(x))+6*x*polylog(2,exp(x))-6*p 
olylog(3,exp(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 875 vs. \(2 (84) = 168\).

Time = 0.10 (sec) , antiderivative size = 875, normalized size of antiderivative = 8.58 \[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\text {Too large to display} \] Input:

integrate(x^3*cosh(x)^2*coth(x)^2,x, algorithm="fricas")
 

Output:

1/32*((4*x^3 - 6*x^2 + 6*x - 3)*cosh(x)^6 + 6*(4*x^3 - 6*x^2 + 6*x - 3)*co 
sh(x)*sinh(x)^5 + (4*x^3 - 6*x^2 + 6*x - 3)*sinh(x)^6 + (12*x^4 - 68*x^3 + 
 6*x^2 - 6*x + 3)*cosh(x)^4 + (12*x^4 - 68*x^3 + 15*(4*x^3 - 6*x^2 + 6*x - 
 3)*cosh(x)^2 + 6*x^2 - 6*x + 3)*sinh(x)^4 + 4*(5*(4*x^3 - 6*x^2 + 6*x - 3 
)*cosh(x)^3 + (12*x^4 - 68*x^3 + 6*x^2 - 6*x + 3)*cosh(x))*sinh(x)^3 + 4*x 
^3 - (12*x^4 + 4*x^3 + 6*x^2 + 6*x + 3)*cosh(x)^2 + (15*(4*x^3 - 6*x^2 + 6 
*x - 3)*cosh(x)^4 - 12*x^4 - 4*x^3 + 6*(12*x^4 - 68*x^3 + 6*x^2 - 6*x + 3) 
*cosh(x)^2 - 6*x^2 - 6*x - 3)*sinh(x)^2 + 6*x^2 + 192*(x*cosh(x)^4 + 4*x*c 
osh(x)*sinh(x)^3 + x*sinh(x)^4 - x*cosh(x)^2 + (6*x*cosh(x)^2 - x)*sinh(x) 
^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*dilog(cosh(x) + sinh(x)) + 192 
*(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sinh(x)^4 - x*cosh(x)^2 + (6*x*c 
osh(x)^2 - x)*sinh(x)^2 + 2*(2*x*cosh(x)^3 - x*cosh(x))*sinh(x))*dilog(-co 
sh(x) - sinh(x)) + 96*(x^2*cosh(x)^4 + 4*x^2*cosh(x)*sinh(x)^3 + x^2*sinh( 
x)^4 - x^2*cosh(x)^2 + (6*x^2*cosh(x)^2 - x^2)*sinh(x)^2 + 2*(2*x^2*cosh(x 
)^3 - x^2*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 96*(x^2*cosh(x)^4 
 + 4*x^2*cosh(x)*sinh(x)^3 + x^2*sinh(x)^4 - x^2*cosh(x)^2 + (6*x^2*cosh(x 
)^2 - x^2)*sinh(x)^2 + 2*(2*x^2*cosh(x)^3 - x^2*cosh(x))*sinh(x))*log(-cos 
h(x) - sinh(x) + 1) - 192*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + ( 
6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cosh(x)^3 - cosh(x))*sinh(x) 
)*polylog(3, cosh(x) + sinh(x)) - 192*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 ...
 

Sympy [F]

\[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\int x^{3} \cosh ^{2}{\left (x \right )} \coth ^{2}{\left (x \right )}\, dx \] Input:

integrate(x**3*cosh(x)**2*coth(x)**2,x)
 

Output:

Integral(x**3*cosh(x)**2*coth(x)**2, x)
 

Maxima [F(-2)]

Exception generated. \[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate(x^3*cosh(x)^2*coth(x)^2,x, algorithm="maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\int { x^{3} \cosh \left (x\right )^{2} \coth \left (x\right )^{2} \,d x } \] Input:

integrate(x^3*cosh(x)^2*coth(x)^2,x, algorithm="giac")
 

Output:

integrate(x^3*cosh(x)^2*coth(x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\int x^3\,{\mathrm {cosh}\left (x\right )}^2\,{\mathrm {coth}\left (x\right )}^2 \,d x \] Input:

int(x^3*cosh(x)^2*coth(x)^2,x)
 

Output:

int(x^3*cosh(x)^2*coth(x)^2, x)
 

Reduce [F]

\[ \int x^3 \cosh ^2(x) \coth ^2(x) \, dx=\frac {39+78 x -36 e^{4 x}-96 e^{4 x} \left (\int \frac {x^{2}}{e^{6 x}-2 e^{4 x}+e^{2 x}}d x \right )+96 e^{2 x} \left (\int \frac {x^{2}}{e^{6 x}-2 e^{4 x}+e^{2 x}}d x \right )-3 e^{6 x}+54 x^{2}+4 x^{3}-6 e^{6 x} x^{2}-4 e^{4 x} x^{3}-68 e^{2 x} x^{3}-30 e^{4 x} x -48 e^{4 x} \left (\int \frac {x}{e^{6 x}-2 e^{4 x}+e^{2 x}}d x \right )+48 e^{2 x} \left (\int \frac {x}{e^{6 x}-2 e^{4 x}+e^{2 x}}d x \right )+4 e^{6 x} x^{3}+12 e^{4 x} x^{4}-12 e^{2 x} x^{4}+6 e^{6 x} x +12 e^{4 x} \mathrm {log}\left (e^{x}-1\right )+12 e^{4 x} \mathrm {log}\left (e^{x}+1\right )+6 e^{4 x} x^{2}-12 e^{2 x} \mathrm {log}\left (e^{x}-1\right )-12 e^{2 x} \mathrm {log}\left (e^{x}+1\right )-102 e^{2 x} x^{2}-78 e^{2 x} x}{32 e^{2 x} \left (e^{2 x}-1\right )} \] Input:

int(x^3*cosh(x)^2*coth(x)^2,x)
 

Output:

(4*e**(6*x)*x**3 - 6*e**(6*x)*x**2 + 6*e**(6*x)*x - 3*e**(6*x) - 96*e**(4* 
x)*int(x**2/(e**(6*x) - 2*e**(4*x) + e**(2*x)),x) - 48*e**(4*x)*int(x/(e** 
(6*x) - 2*e**(4*x) + e**(2*x)),x) + 12*e**(4*x)*log(e**x - 1) + 12*e**(4*x 
)*log(e**x + 1) + 12*e**(4*x)*x**4 - 4*e**(4*x)*x**3 + 6*e**(4*x)*x**2 - 3 
0*e**(4*x)*x - 36*e**(4*x) + 96*e**(2*x)*int(x**2/(e**(6*x) - 2*e**(4*x) + 
 e**(2*x)),x) + 48*e**(2*x)*int(x/(e**(6*x) - 2*e**(4*x) + e**(2*x)),x) - 
12*e**(2*x)*log(e**x - 1) - 12*e**(2*x)*log(e**x + 1) - 12*e**(2*x)*x**4 - 
 68*e**(2*x)*x**3 - 102*e**(2*x)*x**2 - 78*e**(2*x)*x + 4*x**3 + 54*x**2 + 
 78*x + 39)/(32*e**(2*x)*(e**(2*x) - 1))